Crack the Sequence Code: What Is the Term-to-Term Rule for 5, 8, and 11?

What is the term-to-term rule of the following sequence?

5, 8, 11

To solve this problem, we'll follow these steps:

• Step 1: Identify the common difference of the sequence.
• Step 2: Formulate the general expression for the sequence.
• Step 3: Verify the expression with the given sequence terms.

Now, let's work through each step:
Step 1: Start with the sequence $5, 8, 11$. Calculate the difference between consecutive terms: $8 - 5 = 3$ and $11 - 8 = 3$. Hence, the sequence has a common difference of 3.
Step 2: Since the sequence is arithmetic, it can be described using the formula:
$a_n = a + (n-1)d$
where $a = 5$, the first term, and $d = 3$ is the common difference.
Thus, we have:
$a_n = 5 + (n-1) \times 3$
Simplifying further:
$a_n = 5 + 3n - 3 = 3n + 2$
Step 3: Verify this formula by substituting $n = 1$, $n = 2$, and $n = 3$:
For $n = 1$, $a_1 = 3(1) + 2 = 5$.
For $n = 2$, $a_2 = 3(2) + 2 = 8$.
For $n = 3$, $a_3 = 3(3) + 2 = 11$.
Each calculation yields the correct term in the sequence.

Therefore, the solution to the problem is $\mathbf{a_n = 3n + 2}$.